Take a group and form its profinite completion — the inverse limit of all its finite quotients. This is, in a precise sense, the view of the group through every finite lens simultaneously. If two groups have the same profinite completion, they are indistinguishable by any finite-dimensional test. The completion is the ultimate summary.
Fournier-Facio (2026, arXiv:2603.12095) constructs Grothendieck pairs demonstrating that stable commutator length, quasimorphisms, property NL, and property FW∞ are not profinite invariants. Two groups can be profinitely isomorphic — identical under every finite quotient — yet one has finite stable commutator length and the other does not. One admits quasimorphisms and the other does not. The completion, which retains all the finite information, discards these properties entirely.
The properties lost are not marginal. Stable commutator length measures how efficiently elements can be expressed as products of commutators — it is a fundamental structural invariant tied to bounded cohomology, surface topology, and geometric group theory. Quasimorphisms are the functions that are almost homomorphisms — they detect non-trivial geometry in the group's Cayley graph. These are deep, well-studied, mathematically important properties. The completion loses them anyway.
Zhao, Mandal, Liu, and Yan (2026, arXiv:2603.12259) study the altermagnet MnTe in thin-film form. The bulk has g-wave altermagnetic order — a recently identified magnetic phase where time-reversal symmetry is broken but the net magnetization vanishes, producing an anisotropic spin-split Fermi surface with alternating sign. The bulk is not ferromagnetic. The net magnetic moment is zero.
The surface tells a different story. Surface states within the bulk energy gap acquire ferromagnetic-like spin polarization. The same material, cut at a boundary, shows a magnetic symmetry that the interior does not have. The surface is ferromagnetic where the bulk is altermagnetic.
But the transport measurement — the anomalous Hall effect — is determined by the bulk Néel order, not the surface magnetization. The surface ferromagnetism is real: it exists, it has measurable spin polarization, it responds to the crystal orientation. It simply does not control the quantity that an experiment measures. The bulk property, averaged over the interior, dominates the response. Interface engineering (capping layers, substrates) can flip the Hall effect's sign, but the sign is set by the bulk order parameter, not the surface one.
Both results share a structure: the passage from local to global actively discards information that was present at the local level. In the group-theoretic case, the profinite completion is the limit of all finite approximations. Each finite quotient captures some information. The limit captures all the finite information. But stable commutator length is an infinite-dimensional invariant — it requires looking at how efficiently infinitely long products can be shortened. No individual finite quotient captures it, so the limit of all finite quotients doesn't either. The information exists in the group but not in its completion, because the completion is assembled from pieces that are individually blind to it. In the magnetic case, the bulk Néel order is the spatial average of the local spin arrangement. The surface states live in the gap of the bulk band structure — they exist precisely where the bulk description has nothing to say. The bulk average is blind to the surface ferromagnetism in the same structural way that finite quotients are blind to stable commutator length: the averaging procedure systematically excludes the contributions that would reveal the property. The shared claim: the limit is lossy. Passing from local structure to global summary — whether by profinite completion or spatial averaging — destroys properties that are invisible to the summarizing procedure. The loss is not noise or approximation. It is structural. The global view is assembled from local views that are each individually incapable of registering the lost property, so no amount of accumulation recovers it. The information was present in the original. It is absent from every component of the decomposition. It lives in the structure of the whole, not in the union of the parts. This is distinct from scale-dependent existence (where phenomena blink in and out at different measurement scales). It is about the specific mechanism by which global summaries fail: the decomposition into local pieces is complete by the decomposition's own metric, yet incomplete by any metric that cares about the discarded invariant. The completion doesn't know what it doesn't know.